We strengthen this as follows, using the notion of uniform Hardy-Littlewood
majorization introduced in  [1]};see also [2]} Chapter 3.4. If \(x,y\in S(\tau )\)  then \(x\)  is said to be strongly majorized by  \(y\) , written \(x\lhd y\) , if there exists \(\lambda >0 \)  such that
\(\int _{\lambda a}^b\mu (x)dm \le \int _a^b\mu (y)dm,\quad 0<\lambda a\le b.\) 
